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Full text of "Cerenkov radiation from bunched electron beams"

■ id d ac y 

RESEARCH REPORTS DIVISION 
NAVAL POSTGRADUATE SCHOOL 
MONTEREY, CALIFORNIA 93943 

bort Number NPS-61-83-003 



IIBRARY 

f e IS DIVISION 

naval; : ate school 

MONTEREY, CALIFORNIA 9394Q 



NAVAL POSTGRADUATE SCHOOL 

Monterey, California 








CERENKOV RADIATION FROM 






BUNCHED ELECTRON 


BEAMS 


F. 


R. 


Buskirk and J.R. 


Neighbours 






October 19 82 






Revised April 


1983 


Technical 


Report 








Approved for public releaser distribution unlimited 



repared for: 
FEDDOCS hief of Naval Research 

D 208.14/2:NPS-61 -83-003 rlington, Virginia 22217 



NAVAL POSTGRADUATE SCHOOL 
Monterey, California 



Rear Admiral John Ekelund David Schrady 

Superintendent Provost 

The work reported herein was supported by the Chief 
of Naval Research. 

Reproduction of all or part of this report is authorized. 
This report was prepared by: 



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'ittUHlTY CL ASSlFlt ATIONOF Inlt> PtOE iWUm Dmlm tnlmtmj) 



RE PORT DOCUMENTATION PAGE 



1 REPONT NUMBER 

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GOVT ACCESSION NO 



* TITLE (mnd Subitum) 

Cerenkov Radiation from Bunched 
Electron Beams 



7. AUTMO«C«J 



F. R. Buskirk and J. R. Neighbours 



» PERFORMING ORGANIZATION NAME AND AOORESS 

Naval Postgraduate School 
Monterey, CA 93940 



II. CONTROLLING OFFICE NAMC AND AOORESS 

Chief of Naval Research 
Arlington, Virginia 22217 



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5 type of report * perioo coveaeo *, 
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November 19 8 2 



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MONlTOAlNO ACCNCV NAMC A AOORESSflf dlttmtmnt from Controlling Olllcm) 



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Approved for public release; distribution unlimited 



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1*. SUPPLEMENTARY NOTES 



It- KEY WORDS (Contlnum on tmvmtmm mldm II n«C(li«ry «j Idmnllty oy block numbmr) 

Cerenkov Radiation 
Microwave Radiation 
Bunched Electrons 



20. ABSTRACT (Cominum on tmvmtmm mldm It nmemmsmry mnd Idmnllty 6r mloe* m~a>o«<i 

Cerenkov radiation is calculated for electron beams which exceed 
the velocity of radiation in a non dispersive dielectric medium. 
The electron beam is assumed to be bunched as emitted from a 
travelling wave accelerator, and the emission region is assumed to 
be finite. Predictions include (a) emission at harmonics of the 
bunch rate, (b) coherence of radiation at low frequencies (c) smear- 
ing of the emission angle for finite emission regions, (d) explicit 
evaluation of cower sroectrum in terms of bunch dimensions. The 



do ,; 



FORM 
AN 7) 



1473 



EDITION OF I NOV «t IS OBSOLETE 
S/N 102-0 14- 660 1 



UNCLASSIFIED 



SECURITY CLASSIFICATION OF THIS PAGE (Whmn Dmlm Sntmtmdi 



UNCLASSIFIED 



CfaCURITV CLASSIFICATION OF THIS PAGECWi^n Dmtm Entmrmd) 



results apply to microwave emission from fast electrons in 
air or other dielectrics. 



DD Form 1473 (BACK) 
S/N 1 "SfolV-SeO! UNCLASSIFIED 



SECURITY CLASSIFICATION OF THIS PAGE(TWi«n Dmtm Entm 



TABLE OF CONTENTS 

Page 

Introduction 1 

Calculation of Poynting Vector 3 

Fourier Coefficients of the Current 6 

Vector Potential 8 

Radiated Fower 11 

Cerenkov Angle 13 

Discussion of Results 15 

a. Effect of Pulse Size 15 

b. Shearing of the Cerenkov Angle 16 

c. Behavior at high Frequencies Related 16 
to Pulse Parameters. 

Concluding Remarks; 18 

References 20 

APPENDICES 

A. Derivation of Cerenkov Radiation Al 
for a Single Pulse of Charge 

B. Derivation of Equation 7 Bl 

C. Temporal Structure of the Electron CI 
Pulse from a Travelling Wave 

Linear Accelerator 

D. Form Factors Dl 

Distribution List 



I NTRODUCTION 

The radiation produced by gamma rays incident on ordinary 
dielectric materials such as glass was first discovered by 
Cerenkov 1 in 1934 and was described in terms of a charged 
particle (electron) moving faster than light in the medium by 
Frank and Tamm 2 in 1937. A summary of work to 1958 is contained 
in the treatise by Jelly-*. An important application is the 
Cerenkov particle detector which is familiar in any particle 
physics laboratory, and an early and crucial application occurred 
in the discovery of the antiproton 4 - . 

Because the distribution of intensity of Cerenkov radiation 
is proportional to the frequency, the radiation at microwave 
frequencies would be low unless beams are intense and bunched so 
that coherent radiation by many electrons contributes . Danos^ 
in 1955 calculated radiation produced by a planar beam passing 
above a dielectric interface and a hollow cylindrical beam passing 
through a hole in a dielectric. Experimental and theoretical 
investigations at microwave frequencies were reviewed by 
Lashinsky^ in 1961. 

This investigation was motivated by a recent renewed interest 
which has included the study of stimulated Cerenkov radiation, in 
which the electron may be in a medium consisting of a gas' or a 
hollow dielectric resonator^* 9. Recent developments of 
electron accelerators for applications such as free electron 
lasers (FEL) have aimed toward high peak currents in bunches in 
contrast to nuclear and particle physics applications, where low 
peak but high average currents are desirable to avoid saturating 

1 



detectors. The high peak currents in the new accelerators should 
yield enhanced Cerenkov radiation, as is calculated in this 
paper . 



CALCULATION OF THE POYNTING VECTOR 

In the following derivation, we consider the Cerenkov 

radiation produced in a dispersionless medium such as cases or 

other dielectrics, by a series of pulses of electrons such as are 

produced by a traveling wave electron accelerator (Linac). The 

pulses or bunches are periodic, the total emission region is 

finite and the bunches have a finite size. 

In determining the radiated power, the procedure is to 

calculate the Poynting vector from fields which are in turn 

obtained from solutions of the wave equations for the potentials. 

Since the current and charge densities entering into the wave 

equations are expressed in fourier form the resulting fields and 

radiated power also have fourier components. In the derivation, r 

is the coordinate at. which the fields will be calculated, r i.s 

the coordinate of an element of the charge which produces the 

fields and n is a unit vector in the direction of r. We assume 

that E(r,t) and B(r,t,) have been expanded in a fourier series;, 

appropriate for the case where the source current 1.3 periodic. 

Then we have 

00 

E (tt) -£ e" iaJt 1 tf,a») 

oj=-°° (1) 

and a corresponding expansion for B, where w is a discrete 

_& — ^ 
variable and E and B are fourier series coefficients. The 

poynting vector S is given by 



S = -~E x~B 



(2) 



and it is easy to show that the average of S in a direction given 



A 



by a normal vector n i: 

T oo 



i r n.s at - i y. -lu. 

-J /i\=— oo 



oj ) x B(r,-(D) 



0)=-» 



(3) 



where T is an integer multiple of the period of the periodic 
current . 

Letting c = (ye) "1/2 be the velocity of light, in the 
medium, the wave equations for A,4> and their solutions are, 

2 



3 -*■,-*> -*■ » 
V — 9 — \ A(r,t) = yj(r,t) 

c z 3t z i 



I c 2 dt 2 i 



'?ir,t) = 1/e p(r,t) 



m 



3 , 



A(r,t) =y D(r-r', t-t ') J(f',t ') dr 'dt ' 



«f> (r,t) = 1 

e 




D(r-?',t-t , )p(r' ,t')d 3 r'dt' 



(4) 



(5) 



where the Green's function D is given by 



D(r,t) = 1_ 5 (t-r/c) 

4irr 



(6) 



The vector potential A(r, t) also can be developed in a fourier 

series expansion of a form similar to (1) with an expression for 

the fourier series coefficients given by 
T 
A(r,a>) = 1 r dtA(?,t)e 1 ' 



1 f dtA(r,t) 

T I 

■ill 



d 3 r> j(r' r ai)_l 1__ e iaJ l r - r ' ' /c 

r-r' 



(7) 



Now if we assume that the observer is far from the source so 
that jr j>> |r' | for regions where the integrand in (7) is important 
we can let |r-r | = *r - n • r in the exponential and jr - r' = : 
in the |r - r 1 |~1 factor in (7), obtaining (where n = r/r) 

A(r,o>) = Ji_ e^ r/c r[fa 3 r'j;(r' ,a>)e- iiai/crn - t ' 

47rr JJJ (8) 

The fourier series coefficients of the fields are obtained 

from those for the vector potential (8) through the usual 

relations B = V x A and E = -V<|> - 9A . Under the conditions 

3t 
leading to (3) the field fourier coefficients are-^: 

-* - ^ -l ^ 

B(r,uj) = ico n x A ( r , oj ) 

c (9) 

-* -i -A -* -k 

E(r,co) = -c n x B(r,oj) 

(10) 



The poynting vector can now be found by using (9) and (10) 
in expansions like (1) and then substituting in (2). However it 
is more convenient to deal with the frequency components of the 
radiated power by substituting (9) and (10) into the expression of 
the average radiated power (3). 

T 00 



1 [ n-Sdt = 1 ) 
T J y ^— 



oj n x A ( r , oo ) 



c (11) 



F OURIER COMPONENTS OF THE CURRENT 
The expression (7) for the fourier components of the vector 
potential contains the fourier components of the current density. 
Consequently it is necessary to examine the form of the current 
and its fourier development. Assume the current is in the z 
direction and periodic. If the electrons move with velocity v, 
and we ignore for the moment the x and y variables, the charge or 
current functions should have the general form 



f(z,t) = £ e lk z Z £ e ia)t f(k z ,a») (12) 



(13) 



Z CO 

Under the condition of rigid motion, 
f (z,t) = f (z-vt) 

it is easy to show that 



,(k ,cj) = 6 . f (k ) , % 

z co,kv~ z (14) 



where 



fvkj = e 1K z z f (z)dz 

*v o Z — 

z 



(15) 



Thus the restrictions of equation (13) reduce the two 
dimensional fourier series of eq. (12) to essentially a one 
dimensional series (14). 



With (14) in mind, the current density associated with the 
electron beam from a linear accelerator should be periodic in both 
z, t, with a fourier series expansion, but the x and y dependence 
should be represented by a fourier integral form: 



4«o 



J_(r,t) ~ vp(r,t) = 



(2 TT) 



— CO 



dk dk 
x y 



f ,, \~ i (k-r - cat) 
J dk y ^_ e Po 



k =- c 
z 



(k) 
16) 



where the fourier components of the charge density are 



P,o (k) = dx dy - | dze 



U 7 ) 



p (r) is p(r,t) evaluated at t = o and J is assumed to be in 
the z direction. Note in eq. (16) that k z and ^ are both 
discrete and from (14), oj = k z v. 



VECTOR POTENTIAL 

The results of the previous section can be applied to the 
evaluation of the vector potential and in turn to the fields. 

Let the infinite periodic pulse train be made finite, 

extending from z = -Z ' to z = +Z ' and let 9 be the angle between n 

and A. Then the cross product in (11) can be written 

I A *t #"*" \ I a V ioor/c 

n x A(r,oj) = smfl -7 — e 

CO GO 7 ' 



f dx' / dy' /" dz' e " ifi ^ ,a)/c 

Jm J>™ =^>7 I 



JSOO .= 00 — 2 



k =- c 



t-At2 dk / dk ) v0 o (£)<$, e 1K ' r 

(2n) z I x J ot *y k ^ = _^ - k z v /W (18) 



But 



Z* j. * 



dx' J dy' / dz' e ir '- (k ™ /c) 



-Z' 

(2tt) 2 5(k x - n x aj/c) 5 (k - nu/c)I(Z') ( 19 ) 



where 

Z' 

T ,„ n / , , i(k - n co/c)z' •> 

I(Z') = / dz' e z z 2 sinGZ . (20) 

J-Z' G 



and G = k z - n 2 go /c = 00 /v - n z co /c 
And thus the cross product term is 



/\ •*■ ^ 1 Tj icor/c 

n x A(r,a>) I = sinG-^r e v£o(n x Vc, n oi/c, o)/v)I(Z') 



(21) 



Note that. o> is a discrete variable but from 19, the 
continuous variables k x and k y become evaluated at discrete 
points . 

Returning to (17), a more symmetric form may be obtained by 
assuming rhat p (r)# which is periodic in z with period Z, is 
actually zero between the pulses. Denoting by p ' ( r) the charge 
density of a single pulse, which is zero for z < o and z > Z the 
integral on z can be written 

z z 

, -ik z ,-». f , -ik z ',■»> -, -ik z ' ,a 
dz e z p (r; = dz e z p (r) = dz e z p (r) 

J (22) 



Then (17) the fourier coefficient-, of the charge density, becomes 

oo 

p (k) =| |fj r d 3 r e" ik ' r : '(r) = |pl ik) 



(23) 



where p n '(k) is the three dimensional fourier transform of the 
single pulse decribed by p Q ' (r) . 

Substituting these expressions into (21) gives a final simple 
result, for the cross product, form: 

|n x A(r,<o)l = sine H e lajr/C (v/Z) Pe ' (k)I(z') 

47tr - (24) 



where 



I(Z' ) = | sin GZ' 
G 

G = oj/v - n co/c (25 ) 

k = (n oo/cn a)/c,oo/v) 



The components of the Cerenkov E and B fields may now be found by 
substituting (24) in (9) and (10). 



10 



RADIATED POWER 
The frequency components of the average radiated power are 
obtained by substituting (24) into (II). The negative: frequency 
terms equal the corresponding positive frequency terms , yielding a 
factor of 2 when the summation range is changed. Multiplying by 
r^ converts to average power per unit solid angle, dP/dft , 
yielding 

T 00 

f§=r 2 i I" n-Sdt = r 2 l£V |nx A(?, M )| 2 



00 

=1 



C 

W(<o,n) 



(26) 



where w(a)/n) is defined to be 

2 
w (a) ,n) = — 2 g- sin 9 (v /Z ) | p ' (kj j I (Z ) 
(4it) 



(27) 



W(o),n) is the power per unit solid angle radiated at the frequency 
0), which is a harmonic of the basic angular frequency uj of the 
periodic pulse train. 

To find P , the total power radiated at. the frequency co , 
W is multiplied by dfi and integrated over solid angle. Note that 
n z = cos 9, and as 9 varies, G changes according to (25), 



11 



w 



ith dG = - (co/c) dn so that 



d^ = dc{> (c/oj) dG 



(28) 



Noting that the integral over <£ yields 2"^ , we find the result for 
the total radiated power at the frequency oj for all angles 



G" 



p = r_ _ 



2 2 

}J_ 60 v ^ 

4tt c 



2 J 



sin 2 6 jPo(k) | 2 I 2 (Z') 5 dG 



(29) 



G' 



12 



CERENKQV ANGLE 
The remaining integral over G may now be examined. The 
sin^ S and p factors may often be slowly varying compared to 
the l2(z') factor, the latter being shown in Fig. 1. For large 
Z', the peak in l2(z') becomes narrow, and if the integrand may 
be neglected outside the physical range G'<G<G", 



G 

I, 



I (Z ')dG = |4(Z ') 



/ 



i, 2 sinGZ 



'} 



GZ' 



'dG = 4tt 



(30) 



Then, evaluating the sin 9 factor and p Q (k) at the 
point corresponding to G = 0, (which is cos 6 = n z = c/v) shows 
that. 9 at. the peak of I(Z' ) is the usual Cerenkov angle 9 C . We 
thus obtain for large Z' 



P = £- covsin 6 |p (k)| 4ttZ'/Z 

oj 4lT C l ~° ' 



(31) 



Now let 2Z'/Z = ratio of the interaction length to pulse spacing 
N, the number of pulses. Also Z = v2 tt/ o^ or 2tt/z = '^ / v so 
that, (in the large Z' limit), 



2 ** 2 

P oj = 4tt" 000 °o vsin 9 c IP=^ k )i N. 



(32) 



To compare with usual formulations, (32) is divided by Nv to 
obtain the energy loss per unit path length per pulse: 



dE = y_ 
dx 4tt 



o)co sin 8 I p (k) , 



, 1 2 



(33) 



13 



If the pulse is in fact, a point charge, the fourier transform 
Dq (k) reduces to q, the total charge per pulse and (33) is 
very similar to the usual Cerenkov energy loss formula, where for 
a single charge q. the radiation is continuous and the factor 

oo cj in (33) is replaced by u) d go. In the present case the pulse 

o 

train is periodic at angular frequency oj q and the radiation is 
emitted at the harmonic frequenies denoted by oj . 



14 



DISCUSSION OF RESULTS 
Equation (29) and the approximate evaluation expressed as 
(32) form the main results. Some consequences will now be noted. 

a. EFFECT OF PULSE SIZE. The spatial distribution of the 
charge in the pulse appears in p^lk) , which is the fourier 
transform of the charge distribution. The peak of I 2 (2') in 
figure 1 occurs at G = or n z = c/v. Thus at the peak, co/v = n z oj/c 
i\o that k, the argument of p' (k), is evaluated at (from 25) 

s+S 
"*■ A 

k = noo/c 

(34) 

We may also define a charge form factor F(k) 

o;<k) = q p(ki (3S) 

The form factor F(k) is identically one for a point charge, 
and for a finite distribution F(k) = 1 for k =o . 

Furthermore F(k) must fall off as a function of k near the 
origin if all the charge has the same sign. If the pulse were 
spherically symmetric, F(k) would behave as elastic electron 
scattering form factors defined for nuclear charge 
distributions 1-1 . In that case, the mean square radius <r 2 > 
of the charge distribution is given by the behavior of F(k) near 
the origin. 

F(k) ■*■ 1 - <r 2 > k 2 /6 (spherical pulse) (36) 



15 



b. SMEARING OF THE CERENKOV ANGLE. For a finite region over 
which emission is allowed, namely if 2Z ' is finite, the function 
I 2 (Z'), appearing in the integral in (29), will have a finite 
width. Since the peak height is 4Z ' 2 and the area is 4tts ' , 
(30), we can assign an effective width 2T = area/ height - ^ I z ' 



or 



r=7T/2z' (37) 

Thus the radiation is emitted mainly near G = o (which 
corresponds to 9 = 9 C ) hut in a range Ag = +T . But from (25), 

00 , 00 . . . 

AC = — in 7 = — a(cos9) so that there is a range in cos0 over 
c ^ c 

wr. ich emission occurs: 



a / Q^ C 7T (38) 

A (cos9) = - ^—r 

00 ZZj 



Note that the finite angular width of the Cerenkov cone angle 
in (38) has the factor l/co , indicating that the higher harmonics 
are emitted in a sharper cone. 

c. 3EHAVI0R AT HIGH FREQUENCIES RELATED TO PULSE PARAMETERS. 
To be specific let the charge distribution for a single pulse be 
given by gaussian functions 



1 .-* 2 2 ? 2 9 

p (r) = A exp(-x/a - y /a - z7b ) 



Then F(k) may be found 



39) 



F(k) = exp(-k 2 a 2 /4 -k 2 a 2 /4 -k 2 b 2 /4) 

y Z (40) 



Beam pulse parameters could then be determined by measuring the 
Cerenkov radiation. For example, fast electrons from an 
accelerator in air will e ait with a 9 C of several degrees 
in which case k x and k y in (40) can be neglected, giving 

F(k) = exp(-k z 2 b 2 /4) = exp[-co 2 b 2 /(4v 2 ) ] (41) 

The expected behavio:: of P as a function of go is shown 

00 

qualitatively in Fig. 2 as a linear rise at. low frequencies 
followed by a fall off at. higher frequencies, the peak occurring 
at 



(o = v/b 
m 



(42) 



Furthermore, a different behavior would be expected at. very 
high frequencies. The formulation from the beginning represents 
coherent, radiation from all charges, not only from one pulse, but. 
coherence from pulse to pulse. F(k) then describes interference 
of radiation emitted from different, parts of the pulse, but note 
that expressions (29} and (32) will still be proportional to 
q2 - n 2 e 2 w here n ± s the number of electrons in a pulse. 
Thus the n^ dependence of P indicates coherence. But. above 

CO 

some high frequency go j_ such that oj . /c - 2\\ I % , where I is the 

mean spacing of electrons in the cloud, the radition should switch 

over to incoherent radiation from each charge and P should be 

00 

proportional to n. The incoherent radiation should then rise 
again as a function of au . 



17 



CONCLUDING REMARKS 

The general results presented here describe the Cerenkov 
radiation produced by fast electrons produced by a linear 
accelerator. For an S band Linac operating at about 3Ghz (10 cm 
radiation) , the electron bunches are separated by 10 cm and would be 
about 1 cm long at 1% energy resolution. Microwave Cerenkov 
radiation is expected and has been seen in measurements at the Naval 
Postgraduate School Linac. 

Two types of measurements were made. In measurements of Series 
A, an X-band antenna mounted near the beam path, oriented to 
intercept the Cerenkov cone, was connected to a spectrum analyser. 
Harmonics 3 through 7 of the 2.85 GHz bunch frequency were seen but 
power levels could not be measured quantitatively. Harmonics 1 and 2 
were below the wave guide cut off. In the series B measurements, the 
electron beam emerged from the end window of the accelerator, and 
passed through a flat metal sheet 90 cm downstream oriented at an 
angle <j> from the normal to the beam. The metal sheet defined a 
finite length of gas radiator, and reflected the Cerenkov cone of 
radiation toward the accelerator but rotated by an angle 2<t> from the 
beam line. A microwave X-band antenna and crystal detector with 
response from 7 to above 12 GHz could be moved across the (reflected) 
Cerenkov cone as a probe. 

As mentioned earlier, the series A measurements showed the 
radiation is produced at the bunch repetition rate and its harmonics. 
Series B measurements performed with several antennas always 
indicated a broadened Cerenkov cone with strong radiation occuring at 
angles up to 10°, well beyond the predicted Cerenkov angle of 1.3°. 



18 



Since a broad band detector was used it was impossible to verify 
the prediction (see eq . 38) that the broadening cf the cone should 
depend on the harmonic number. However, it should be noted that 
incoherent radiation by a beam of lu A at 9 c = 1.3° for a 1 
meter path in air would be about 10~^ watts at microwave 
frequencies so that observation of any signal by either method A 
or B clearly demonstrated coherent radiation by the electron 
bunches . 

Many of the concepts were clearly notea by Jelly in his 
treatise (Jelly^, Section 3.4 especially). The form factor was 
noted but a general expression was not given. In fact, the form 
factor quoted by Jelly represents the special case of a uniform 
line charge of length L' with a projected length L=L'cos9 c in 
the direction of the radiation. The coherence of the radiation 
from the bunch was noted but no broadening of the: cone nor 
harmonic structure were developed. 

Casey, Yen and Kaprielian-^ considered an apparently 
related problem in Cerenkov radiation, in whj ch a single electron 
passes through a dielectric medium, where a spatially periodic 
term is added to the dielectric constant. The result is radiation 
occurring even for electrons which do not exceed the velocity of 
light in the medium, and at angles other than the Cerenkov cone 
angle. The non-Cerenkov part of the radiation is attributed to 
transition radiation. 

In the present paper, the transition radiation associated 
with the gas cell boundaries is included, and radiation appears 
outside the Cerenkov cone. 



19 



If the electron velocity were lower so that v/c were close to but 
less than unity, the peak in I would be pushed to the left in Fig. 
1, such that cos 9 c = v/c would be larger that 1. But. the tails of 
the diffraction function I would extend into the physical range 1 _< 
cos 8 <_ - 1 , and this would be called transition radiation and he 
ascribed to the passage of the electrons through the boundaries of 
the gas cell. Now return to the case v/c > i, with the situation as 
shown in Fig. 1. The radiation is then a combination of Cerenkov and 
transition radiation. The formalism of Reference 12 does admit a 
decomposition into the two types of radiation, but is inherently much 
more cumbersome. 

As a final remark, one might extend the analysis further in the 
region near w ^ . Consider electron bunches emitted frcm a 
travelling wave Linac, which could be 1 cm long spaced 10 cm apart. 
Let these bunches enter the wiggler magnet of a free electron laser 
(FEL) . Then, if gain occurs, the 1 cm bunches would be subdivided 
into bunches of a finer scale, with the spatial r-cale appropriate to 
the output wavelength of the FEL. 13 If the (partially) bunched 
beam from the FEL were passed into a gas Cerenkov cell, then the 
observed radiation should be reinforced because cf partial coherence, 
at the FEL bunch frequency and harmonics. This would lead to bumps 
in the spectrum in the region near oj ^ . 



20 



ACKNOWLEDGEMENTS 

This work was made possible by the Office of Naval Research 
W. B. Zeleny provided much insight into the theory for point 
charges, and the experiments performed by Ahmet Saglern, Lt . , 
Turkish Navy, provide preliminary substantiation of the work. 



21 



REFERENCES 

1. Cerenkov, P., Dcklady Akad-Nauk, S.S.S.R., 2 451, (1934). 

2. Frank, I.M. and Tamm, I., Doklady Akad-Nauk 3.S.S.R., L4 109, 
(1937) . 

3. Jelly, J. V., "Cerenkov Radiation and its Applications". 
Pergammon Press, London, (1958). 

4. Chamberlain, 0., E. Segre, C. Wiegand and T. Ypsilantis, Phys . 
Rev. , 100 947, (1955) . 

5. E'anos, M.J., J. Appl. Phys, 26_ 1 , (19 55). 

6. Laskinsky H. "Cerenkov Radiation at Microwave Frequencies" 
in advances in Electronics and Electron Physics, L. Marton, 
(ed) Academic Press, New York, Vol. XIV, pp. 265, (1961). 

7. Piestrup, M.A., R.A. Powell, G.B. Rothbart, C.K. Chen and R.H. 
Fantell, Appl. Phys. Lett., 28. 92, (1976). 

S. Vialsh, J.S., T.C. Marshall and 3. P. Schlesinger, Phvs . Fluids, 
20 709 1977. 

9. Valsh, J.E. "Cerenkov and Cerenkov-Raman Radiation Sources" in 
P hysics of Quantum Electronics , S. Jacobs, M. Sargent and 

M. Scully (ed) Vol 7, Addison Wesley, r eading, Mass., (1979). 

10. Jackson, J.D., "Classical Electrodynamics", (2 nc ^ Edition) 
John Wiley, New York 1975 (pp.4) Sec. 9.2 pp. 394ff. 

11. Hofstadter , R. , Ann. Rev. Nuc Sci, Vol 7. pp. 2 31, (1937). 

12. Casey, K.F., C. Yeh and Z.A. Keprielian, Phvs. Rev. 140 B 768 
(1965) . 

13. Hopf, F.A., P. Meystre and M.O. Scully, Phys. Rev. Lett 3_7 
1215 (1976) . 



22 




r=TT/2Z 



J! 



*G 



* e 



Tf 



^i 



igure 1. 



Qualitative Behavior of the Function I 2 (Z'). Both 
the function G, from Eq. 25 in the text, and the 
emission angle are displayed as independent 
variables. G' and G" are upper and lower limits. 



23 



Pu 




» OJ 



Fiqure 2. Schematic Behavior of Power Emitted as a function of 



Angular Frequency 



24 






F 3 Eo 



o - 




— r 



♦2 AY 



W 




o 



i — i — 

2TT 



-5- ^; 



Figure 3. Structure of charge pulse from a travelling wave 
accelerator. y is the phase angle of an electron 
relative to the peak of the travelling wave 
accelerating field. Electrons in the range +A^are 
passed by magnetic deflection system. 



25 



APPENDIX A 
DERIVATION OF CERENKOV RADIATION FOR A SINGLE PULSE OF 
CHARGE . 

Let the pulse be described by 

p' (r,t) = p ' Q (r- vt) Al 

Both k z and go are continuous variables in this case; "v is again 
along the z axis. If we expand in terras of a four dimensional 
fourier integral, 

p'(r,t) = l/(2ir) 4 e 1 ^ " k - r ) o ' (k, w) d 3 kdoj A2 

It may be shown that the condition Ai gives: 



p'(k,aj) = 2tt6(o) - k z v) p' Q (k) A3 



where p' (k) is the three dimensional spatial tranform of p 1 
evaluated at t = o. All the fields have fourier integral rather 
than fouries series expansions and the energy radiated per unit 
solid angle become 



2 ^ ~ £ X ] - -, u f 2, 

dt n • S = t" ,,,, 2 — (o dw 
2tt (4tt) c J _ 



'Jd 3 r' J df e i(ct'-n.?')o)/c. x ?( *, tl) | 



A4 
m |2 



W ("> , n ) do) 



Al 



The integrand is a symmetric function of co so that 



W 



<-'*-^3*"\ffl) r * w 



icu(t' -n-r' /c) ^ £,«»■, . , , 
e nxJ(r ! ,t') 



A5 



where 



1 y ,* -, 2.,2 

= — -, — co vn x v) M 

, - 3 c 

lOTT 



M = 




,3 , , Mcot'-n-r /c) ,,-"", ,, \ 
d r d-c e p (r , t ) 



A6 



Now we may write p'(r',t') in a fourier integral representation 



•(r\t') 



(2ir) 



J. 



*3. , , , ,,.-*, M -i(o) 't'-k' -r ') 

a k dco p ( .< , co ) e 



A7 



Inserting eq . A3 into eq. A7 and the result into eq A6, the 
integral over d^ ' involves only exponentials and yields 
(2rr)3 o3(y v ' _ om/c) , so that eq A6 becomes 



II d 3 k'dco'e i(a) " a) " )t, 6 3 (k' - ncu/c: 



5 (aj'-k. 'v)p' (k') 



Now the integral over go' may be done; because of the 6 function, 



to' is evaluated at k z 'v. 



M = 



/dt^AV 



i i 



L(co-k v) t £.3.^, a. . .,,-*.. 

z (S (k ' - con/c) p ' (k ' ) 

-v o 



A2 



Now do the integrals over k x ' , k y ' and k z ', noting that 
k ' appears in the exponential, but k ' and k ' do not, 



u _ [,, icot' -iwt'n 
M = / dt e e : 



v / c ' / / / / s 

; p (am /c,am /C/um /c) 
~ x y z 



This may be written as 



M = dt e p (oon/c) A8 



where 



H = i-n v/c 

z A9 



If we let the time interval be finite, from -T to + T, the 
integral is easily done: 



2_ 



iM = — sin ojHTPo (nco/c) A10 



M = 4T sin cjHt I Po (noo/c) | A ]_l 

(coHT) 2 



A3 



This result, eq. All may be inserted into A5 for oo • The 
factor n x v is just sin 9 where 9 is the angle between the 
radiation and the beam axis. 



W(oj, n ) = —- — -H- (a sin Z 9 4T Z sin coHT' l p ' (nca/c) | Al3 

16 * C (wHT) 2 ' 



W is the energy radiated per unit solid angle per unit 
angular frequency, co • To proceed to the total energy, multiply 
by d ft (solid angle) and integrate. But n z = cos 9 so that dQ 
may be related to cH : 



d ft = d(cos 8)dft= - — dHdft A , ,, 

v A14 



Tii'i functions in eq . A13 do not contain <j> so that integration 
over G yields 2 it. Thus: 

wi , j n 1 y 2 m 2f . 2 , ',2 sin coHT ,„ 

A(co,nidft = — co T /sin 9|p | dH 5 

2rr 2 ~ (ooHT) 2 

The sm 2 goHT / (coHT) 2 factor in the integral is peaked at H = 0, 

c 
which by eq. A9 is at n z = cos 9 = — , or the usual Cerenkov 

angle, 9 c . This function is more strongly peaked about H = 

for large values of T, and in fact, for large T we may evaluate 

sin^ and P ' at the point corresponding to H =0. Then the 

integral 



oo 



J 



2 2 

dx sin (ax) /(ax) = rr/a 



A4 



may be used to evaluate eq . A15, yielding 
ffwdfi = i- -| 2Tsin 2 9c|p;(Vc)| 2 



JJ 



A16 



The emmission was assumed to occur in a time interval from -T to 
•KT ; accordingly dividing by 2T yields a rate of emmission, and 
multiplying by v converts to emmission per unit path length. Thus 
we obtain, for the large T limit: 



dxdco 



GO) = 



— ujdco sin 8 |o'(nco/c)| 



A17 



where d^E/^^ is the energy emitted per unit path length 
per unit angular frequency range co . 

The corresponding expression for T not. large is 

H" 



d 2 E , U , /coT 

-= — -=— dco = -: — go dco — 
dxdco 4tt it 



sin 9jp * (noj/c) J sin^coHT 
H' (coHT) 



Aia 



where H" and H' are the value of H corresponding to 9=0 and 
5 = tt respectively. 

Equations Al 7 and A18 then describe the energy radiated per 
unit path length and per unit angular frequency range. For the 
non periodic (single) pulse the radiation has a continuous 
freauency spectrum. For a point charae q, p' (k) is identically q 
and the usual Cerenkov formula is obtained. Equation A17 is 
quoted by Jelly, but only with the form factor corresponding to a 
uniform line charge of length L' . 



APPENDIX B 
DERIVATION OF EQUATION 7 

Equation 7 is derived for the case in which J(r,t) is 

— » 
expanded in fourier series. Let the fourier coefficient for A 

be given by: 

T 

r 

A(r\o)) = i J dt A(r\t) e icut B1 

o 

Assume that the green's function solution for A(r,t) is given: 



(r,t) = uj r /Td 3 r' | 



dt' J(r',t) D(r - r', t - t) 



32 



where 



D(r,t) - ~ Sit - r/c) 



Let the current density be expanded in a fourier series 



B3 



J(?\f ) =Z e- iw ' t, J(?',a J ) B4 



co' 



Then insert 32, B3 and B4 into 31 to obtain 

t (*.«> = H. J dt e 1 ^ JjfdV Jc.f ^ ,^— p- 



B5 



e ia),t ' J(r« ,03') 



_> 



z 



Bl 



Do 
-ioi 't' 



Jl dt ' , note that t' appears in the <5 function and in 
The result is t' is evaluated at t' = t - j r-r 1 l/c. 

T 



u 



A(r,co) = — 

~ T 



dt e 



loot 




d J r' 



1 1 



4tt i-» •»' 

r-r 



B6 



I 



0) 



-ico 't ico ' j r-r ' I /c ^, -^, 1, 
e e ' ' J(r ,a> ) 



Do the integral on t, note that 

T 



kj dt •" 



L (co-co ') L 



= 6 



CO CO 



B7 



Then do the sum on co ' 



A(r ,co; = 



4tt 



d r 



J(r ,co; e 



ico I r-r 1 /c 



r-r 



38 



This proves the desired result, B8 is equation 7 as used in 
the main text. 



Appendix C 

TEMPORAL STRUCTURE OF THE ELECTRON PULSE FROM A TRAVELLING 
WAVE ACCELERATOR. 

Assume that the energy of a single electron emerging from a 
linac with phase $ relative to the travelling wave field is 



CI 



E = E cos-J; 
o 



This relation is shown on fig. 3, along with some dots 
representing electrons rear the maximum energy E c , with phases 
clustered about 'b = o and ty - 2^. Two bunches, separated by a 
phase difference of 2t, are separated by a time T-i = 1/f where 
f is the accelerator frequency, which is f Q = 2.85 x 10^ Hz 
for a typical S-band accelerator of the Stanford type. 

If a deflection system with energy resolution slit passes 
only energies E from E Q to E Q - E the corresponding range of phase 

Aip is 

C2 

AE = E - E = E (1- cosAijj) 
o c 

For AiJj small, this reduces to 



AE = (Ai|;) 2 c3 

E o " 2 



CI 



The temporal pulse length T2 is 



T n = 2A<|; T,/2 

C4 



2 x 



or 



T_ = T • 2u^/2tt 



If C3 is used to evaluate Aii> in terms of the fractional 
energy resolution AE/E Q 

O 7T CD 



For 1% energy resolution, 1 , 2 /T 1 is about 1/20. The 
electrons thus emerge in short bunches, and the charge and 
current, when expressed in a fourier expansion, should have very 
strong harmonic content up to and aoove the 20th harmonic. 



C2. 



Appendix D 
FORM FACTORS 

This section provides details and examples of form factors 
for various charge distributions. From the main text, F differs 
only from P, the fourier transform of P, by the total charge q of 
the bunch, so that for k = o, F reduces to unity. Thus we define 



£ /// d3r P(r)e i ^'" 



F(k) = £ / M d J r Pirje^ 1 L 

q 

For spherically symmetric charge distributions , let k*r = kru, 
where u is the cosine of the angle between k and r. In spherical 
coordinates, d^r = dudyr 2 dr. Then we find, 

CO 



F ( k ) =1±I / dr r p ( r ) 

r~< 1, -J 



sin Kr D2 

q k 

For k very small, sin x may be replaced by x - x 3 /6 and we 
have 

*° 

F(k) = 1 ±1 / dr r p(r)[kr - k 3 r 3 /6] D 3 

q k J 

Then the two terms in the square bracket lead to separate 
integrals, the first term being unity and the second is similar to 
the integral used to calculate the mean square radius, <r~>, 
exceot for a factor k 2 /6. Thus we have 



F(k) = 1 - k 2 <r 2 >/6 



D\ 



D4 



For a uniform spherical charge distribution of radius R, as 
well a a spherical shell of radius k, the integral D2 may be 
performed easily 



F(k) = 3 (sin kR - kR cos kR) 

(Solid sphere) 



(kR) T 



D5 



F(k) = I sin(kR) (Spherical shell) 
kR 



D6 



For a line charge concentrated on the z axis, we may return 
to Dl and let p(r) = 5(x) <5(y) p"(z), so that 



F(k) = 1 dz p"(z) e ikz (line charge) 



D7 



Q 



F(k) - j sin ( _) (uniform line charge 

kZ 2 

of length Z) 

D8 



Distorted spherical symmetry may be said to occur if the 
scale transformation z' = pz serves to make p spherically 
symmetric in tine prime system. Let F g be the form factor 
calculated by 2 in the prime frame. It is simple to show that 

F(k x , k y , k z ) = F s (k^, k^, k^/p) D9 



DZ 



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